Satellite Navigation error sources and position estimation

Transcription

1 Satellite Navigation error sources and position estimation Picture: ESA AE4E08 Sandra Verhagen Course , lecture 6 1

2 Today s topics Recap: GPS measurements and error sources Signal propagation errors: troposphere Multipath Position estimation Book: Sections ,

3 Recap: error sources satellite: orbit clock instrumental delays signal path ionosphere troposphere multipath receiver clock instrumental delays other spoofing interference 3

4 Recap: Code and Carrier Phase measurements f 2 1 s ρli = r+ I 2 L1 + T + c δtu δt ε ρ f + i f Φ = + + δ δ + λ + ε Φ 2 Li r 1 s I 2 L1 T c tu t f i ALi Li Li 4

5 Signal propagation errors: ionosphere ionosphere ς IP : ionospheric pierce point ς Earth IP h I h I : mean ionosphere height RE sinς = sinς R + h E I R E I 1 ( ς = cosς I z 5

6 Signal propagation errors: ionosphere 1 cosς obliquity factor zenith angle ς 6

7 Signal propagation errors: ionosphere zenith delay mid-latitudes: 1-3 m at night 5-15 m mid-afternoon peak solar cycle near equator: max. ~36 m 7

8 Signal propagation errors: ionosphere How to deal with ionosphere? apply ionosphere-free combination (dual-frequency receiver required apply model (reduction 50 70% relative positioning (later this course

9 Signal propagation errors: troposphere 9 km (poles 16 km (equator Dry gases and water vapor Recall: non-dispersive, i.e. refraction does not depend on frequency Propagation speed lower than in free space: apparent range is longer (~ m Same phase and group velocities T ρ = T = T = T = L 1 ρl 2 φl1 φl 2 T 9

10 Signal propagation errors: troposphere Refractivity N = ( n R 6 [ ] Δ ρ = n( l 1dl = 10 N( l dl S R S N = N T = 10 d 6 + N w N( l dl = 10 6 [ Nd ( l + N w( l ] dl = Td + Tw N N d w P T e T 2 P : total pressure [mbar] T : temperature [K] e : partial pressure water vapor [mbar] if known refractivity known 10

11 Signal propagation errors: troposphere Satellite tropospheric delay computed hydrostatic delay T = md ( el* Tz, d + mw( el * Tz, w mapping functions Figure: H. van der Marel Earth Receiver Unknown tropospheric zenith wet delay 11 11

12 Signal propagation errors: troposphere Saastamoinen model: zenith dry and wet delays calculated from temperature, pressure and humidity (measurements or standard atmosphere, height and latitude Hopfield model: dry and wet refractivities calculated Dry delay in zenith direction m at sea level can be predicted with accuracy of few mm s Wet delay depends on water vapor profile along path, 0 80 cm accuracy of models few cm s If no actual meteorological observations available (standard atmosphere applied: total zenith delay error 5 10 cm 12

13 Signal propagation errors: summary ionosphere troposphere height km 0 16 km variability diurnal, seasonal, solar cycle (11 yr, solar flares low zenith delay meters tens of meters m (sea level el=30 o obliquity factor el=15 o el= 3 o 3 10 modeling error (zenith 1 - >10 m 5 10 cm (no met. data dispersive yes no all values are approximate, depending on location and circumstances

14 Signal propagation errors Homework exercise: make plots of the different mapping functions (page 173 Misra and Enge as function of the elevation angle (ranging from 0 90 o compare them with each other AND with the obliquity factor of the ionosphere delay (slide 22 try to explain the differences more details: see assignment on blackboard 14

15 Multipath Signal reflected: arrives via two or more paths at the antenna Reflected signals have different path length and interfere with direct signal Systematic error (does not average out pseudorange error: up to tens of meters carrier phase error: up to 5 cm

16 Multipath direct signal reflected signal phase shift 180 o Figure: H. van der Marel

17 Multipath direct signal reflected signal phase shift 180 o Figure: H. van der Marel

18 Multipath Primary defense: careful selection of antenna locations (away from reflectors not always possible carefully designed antennas (choke rings; microstrip no signals from below signal processing (correlators Pseudorange multipath can be detected/analyzed by forming a special linear combination of code and carrier phase data For GPS: if receiver is static, same multipath pattern repeats after 23h56m (same orbit

19 Multipath: example Mc = C *L *L2 [m] Figure: H. van der Marel multipath [m] multipath on C/A-code pseudorange MC PRN28 PRN28 between 7 and +6 m! April 2 nd, :10-14:30 UT time [sec.] time [s]

20 GPS error budget Empirical values, actual values depend on receiver, atmosphere models, time and location Error source RMS range error [m] satellite clock and ephemeris σ RE / CS = 3 m = SIS URE atmospheric propagation modeling receiver noise and multipath σ RE / P = 5 σ RE / RNM = 1 m m User Range Error (URE σ URE = 6 m σ = σ + σ + σ URE RE / CS RE / P RE / RNM

21 Carrier-smoothing ( s Φ ( ti = r( ti + c δtu( ti δt ( ti τi + T( ti I( ti + λ A+ ε Φ ( ti = ρ precise estimate for change in pseudorange: ΔΦ ( t = Φ( t Φ ( t = Δρ ( t Δ I( t + Δε ( t i i i 1 IF i i Φ i IF ( t i 1 M 1 ρ( ti = ρ( ti + ρ( ti 1 +ΔΦ( ti M M ρ( t = ρ( t 1 1 [ ] near zero if epochs close together carrier-smoothed pseudorange = weighted average of pseudorange (code and carrier-derived pseudorange 21

22 22 Non-linear observation equations ( ( ( x x = + + = ( 2 ( 2 ( 2 ( ( k k k k k y y y y x x r [ ] ( ( ( ( ( ( k k u k k k k t t c T I r ρ ε δ δ ρ =

23 Non-linear observation equations Corrected pseudorange: account for satellite clock offset, and compensate remaining error sources Note: increased noise, since corrections/models are not perfect b ρ = r + cδt + % ε = x x + b+ % ε ( k ( k ( k ( k ( k u ρ ρ r ( k = = ( ( k 2 ( ( k ( ( k x x y y y y ( k x x 2 23

24 Linearization non-linear model Taylor series y = H(v + ε H(v 0 y = H(v 0 + ( v v ε v observed-minuscomputed observations correction to approximate values design matrix δ y = y H(v δ v = v v 0 H(v 0 A = v 0 linearized model: δ y = Aδ v+ ε 24

25 y = ρ = x x + b + % ε ( k ( k ( k ρ H( v = ρ = x x + b ( k ( k Linearization H(v 0 y = H(v 0 + ( v v ε v Approximations required for: satellite position at time of transmission t - τ (from ephemeris problem: τ not precisely known receiver position at time of reception t receiver clock error 25

26 y = ρ = x x + b + % ε ( k ( k ( k ρ H( v = ρ = x x + b ( k ( k Linearization T ( k T H(v 0 x x 0 ( k x x x0 H(v y y H(v ( 1 0 ( k 0 y 0 ( k ( k T = = x x0 = 1 v H(v 0 ( k z z z0 ( k x x0 H(v 0 b 1 H(v 0 y = H(v 0 + ( v v ε v 26

27 Linearized code observation equations δρ (1 (1 T δρ ( 1 1 (2 (2 T δρ ( 1 1 δ x = = δb + ε% M M M ( K ( K T δρ ( 1 1 ρ G 27

28 Least-squares estimation y = Ax + ε; Q yy 1 yy Q linear model ( T 1 x A Q A T 1 = A Q y variance matrix ˆ xx ˆ ˆ yy 28

29 linearized model Least-squares estimation δρ = Gδ v+ ε% Q ; ρρ iteration required, Gauss-Newton method: v 0 =... while Q vv ˆˆ δvˆ 2 Q vv ˆˆ δρ= ρ H(v G = = H(v 0 v η ( T 1 GQ ρρg 1 T 1 δ vˆ = Qvv ˆˆ G Qρρδρ vˆ = v + δ ˆ 0 v v = ˆ 0 v end 0 29

30 Summary and outlook GPS measurements and error sources Linearized observation equations position estimation Next: Position, Velocity and Time (PVT estimation 30